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Factors of 156: Prime Factorization, Methods, and Examples
The factors of 156 are the numbers that completely divide 156 and yield zero as a remainder. Other than that, these divisors produce a whole number quotient. Both these divisors and whole number quotients are called factors.
Since the number 156 is even composite so it consists of multiple factors. In this article, we will take a detailed overview of all these factors and how to determine them.
Factors of 156
Here are the factors of number 156.
Factors of 156: 1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156
Negative Factors of 156
The negative factors of 156 are similar to its positive factors, just with a negative sign.
Negative Factors of 156: -1, -2, -3, -4, -6, -12, -13, -26, -39, -52, -78 and -156
Prime Factorization of 156
The prime factorization of 156 is the way of expressing its prime factors in the product form.
Prime Factorization = 2$^{3}$ x 3 x 13
In this article, we will learn about the factors of 156 and how to find them using various techniques such as upside-down division, prime factorization, and factor tree.
What Are the Factors of 156?
The factors of 156 are 1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, and 156. All of these numbers are the factors as they do not leave any remainder when divided by 156.
The factors of 156 are classified as prime numbers and composite numbers. The prime factors of the number 156 can be determined using the technique of prime factorization.
How To Find the Factors of 156?
You can find the factors of 156 by using the rules of divisibility. The rule of divisibility states that any number when divided by any other natural number then it is said to be divisible by the number if the quotient is the whole number and the resulting remainder is zero.
To find the factors of 156, create a list containing the numbers that are exactly divisible by 156 with zero remainders. One important thing to note is that 1 and 156 are 156’s factors as every natural number has 1 and the number itself as its factor.
1 is also called the universal factor of every number. The factors of 156 are determined as follows:
\[\dfrac{156}{1} = 156\]
\[\dfrac{156}{2} = 78\]
\[\dfrac{156}{3} = 52\]
\[\dfrac{156}{4} = 39\]
\[\dfrac{156}{6} = 26 \]
\[\dfrac{156}{12} = 13\]
\[\dfrac{156}{13} = 12 \]
\[\dfrac{156}{26} = 6 \]
\[\dfrac{156}{39} =4\]
\[\dfrac{156}{52} = 3\]
\[\dfrac{156}{78} = 2\]
\[\dfrac{156}{156} = 1\]
Therefore, 1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, and 156 are the factors of 156.
Total Number of Factors of 156
For 156 there are 12 positive factors and 12 negative ones. So in total, there are 24 factors of 156.
To find the total number of factors of the given number, follow the procedure mentioned below:
- Find the factorization of the given number.
- Demonstrate the prime factorization of the number in the form of exponent form.
- Add 1 to each of the exponents of the prime factor.
- Now, multiply the resulting exponents together. This obtained product is equivalent to the total number of factors of the given number.
By following this procedure the total number of factors of 156 is given as:
Factorization = 1 x 2$^{2}$ x 3 x 13
The exponent of 1, 3, and 13 is 1., where 2 has an exponent of 2.
Adding 1 to each and multiplying them together results in 24.
Therefore, the total number of factors of 156 is 24, where 12 are positive factors and 12 are negative factors.
Important Notes
Here are some important points that must be considered while finding the factors of any given number:
- The factor of any given number must be a whole number.
- The factors of the number cannot be in the form of decimals or fractions.
- Factors can be positive as well as negative.
- Negative factors are the additive inverse of the positive factors of a given number.
- The factor of a number cannot be greater than that number.
- Every even number has 2 as its prime factor which is the smallest prime factor.
Factors of 156 by Prime Factorization
The number 156 is composite. Prime factorization is a useful technique for finding the number’s prime factors and expressing the number as the product of its prime factors.
Before finding the factors of 156 using prime factorization, let us find out what prime factors are. Prime factors are the factors of any given number that are only divisible by 1 and themselves.
To start the prime factorization of 156, start dividing by its smallest prime factor. First, determine that the given number is either even or odd. If it is an even number, then 2 will be the smallest prime factor.
Continue splitting the quotient obtained until 1 is received as the quotient. The prime factorization of 156 can be expressed as:
156 = 2$^{2}$ x 3 x 13
Factors of 156 in Pairs
The factor pairs are the duplet of numbers that when multiplied together result in the factorized number. Depending upon the total number of factors of the given numbers, factor pairs can be more than one.
For 156, the factor pairs can be found as:
1 x 156 = 156
2 x 78 = 156
3 x 52 = 156
4 x 39 = 156
6 x 26 = 156
12 x 13 = 156
The possible factor pairs of 156 are given as (1, 156), (2, 78), (3, 52), (4, 39), (6, 26), and (12, 13).
All these numbers in pairs, when multiplied, give 156 as the product.
The negative factor pairs of 156 are given as:
-1 x -156 = 156
-2 x -78 = 156
-3 x -52 = 156
-4 x -39 = 156
-6 x -26 = 156
-12 x -13 = 156
It is important to note that in negative factor pairs, the minus sign has been multiplied by the minus sign due to which the resulting product is the original positive number. Therefore, -1, -2, -3, -4, -6, -12, -13, -26, -39, -52, -78, and -156 are called negative factors of 156.
The list of all the factors of 156 including positive as well as negative numbers is given below.
Factor list of 156: 1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 12, -12, 13, -13, 26, -26, 39, -39, 52, -52, 78, -78, 156, and -156
Factors of 156 Solved Examples
To better understand the concept of factors, let’s solve some examples.
Example 1
How many factors of 156 are there?
Solution
The total number of Factors of 156 is 12.
Factors of 156 are 1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, and 156.
Example 2
Find the factors of 156 using prime factorization.
Solution
The prime factorization of 156 is given as:
156 $\div$ 2 = 78
78 $\div$ 2 = 39
39 $\div$ 3 = 13
13 $\div$ 13 =1
So the prime factorization of 156 can be written as:
2$^{2}$ x 3 x 13 = 156